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Full text of "Mathematical And Physical Papers - Iii"

ON THE MOTION OF PENDULUMS.                       19
the same size as those employed by Bessel, oscillating once in a second nearly.
The discussion of the experiments of Baily and Bessel belongs to Part II. of this paper. They are merely briefly noticed here to shew that some results of considerable importance follow readily from the general equations, even without obtaining any solution of them.
7. Before proceeding to the problems which mainly occupy this paper, it may be well to exhibit the solution of equations (2) and (3) in the extremely simple case of an oscillating plane.
Conceive a physical plane, which is regarded as infinite, to be situated in an unlimited mass of fluidrand to be performing small oscillations in the direction of a fixed line in the plane. Let a fixed plane coinciding with the moving plane be taken for the plane of yzt the axis of y being parallel to the direction of motion, and consider only the portion of fluid which lies on the positive side of the plane of yz. In the present case, we must evidently have u = 0, w = 0 ; and py v will be functions of x and t, which have to be determined. The equation (3) is satisfied identically, and we get from (2), putting //,=////?,
^ = '    dT     2? ......................     '
The first of these equations gives p = a constant, for it evidently cannot be a function of tt since the effect of the motion vanishes at an infinite distance from the plane; and if we include this constant in II, we shall havejp = 0. Let Fbe the velocity of the plane itself, and suppose
F=csinwtf ........................... (9).
Putting in the second of equations (8)
v = Xl sin nt + X^ cos nt ................... (10),
,d*X,       Y        .ffX,       ftfX,
we get     nK, = p -^,  nZ, = -/* ~^2 =~-^
The last of these equations gives
sin        x + 5cos J      x)
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